EXERCISE 5. Circuit dynamics

1. EXERCISE 5.Circuit dynamics Circuit Theory PoliTong – A.A. 2010-2011 M. Repetto, Dipartimento di Ingegneria Elettrica - Politecnico di Torino S. Leva, Dipartimento di Energia – Politecnico di Milano

2. I.1. Exercise. E1.1 The switch in the network shows in figure is close, in t=0 it is opened. Find the voltage across R1 from 0- to + . The voltage across R1 is a network variable so that the networks in 0- , 0+ , + and the time constant must be calculated. The network in 0-, before the switch is moved, is in DC steady-state. The value of the variables in 0- are calculate solving the network in which inductor is substitute by short-circuit (see figure). We have to calculate the voltage v and the state variable iL.

3. I.1. Exercise. The values of the network variables in 0⁺ are calculate solving the network in which, in the new position of the switch, the inductor is substituted by a current source.

4. I.1. Exercise. In the new steady-state condition at +, the value of the variables are calculate solving the network in which inductor is substitute by short-circuit. The time constant is calculated applying the equation: The value of the equivalent resistance is the one at the terminals of L of the network where all sources have been zeroed.

5. I.1. Exercise.

6. I.1. Exercise. E1.2 The switch in the network shows in figure is open, in t=0 it is closed. Find the voltage across R1 from 0- to + . The voltage across R1 is a network variable so that the networks in 0- , 0+ , + and the time constant must be calculated. The network in 0-, before the switch is moved, is in DC steady-state. The value of the variables in 0- are calculate solving the network in which capacitor is substitute by open circuit (see figure). We have to calculate the voltage v and the state variable vC.

7. I.1. Exercise. The values of the variables in 0⁺ are calculate solving the network in which, in the new position of the switch, the capacitor is substituted by a voltage source. In the new steady-state condition at +, the value of the variables are calculate solving the network in which capacitor is substitute by open-circuit.

8. I.1. Exercise. The time constant is calculated applying the equation: The value of the equivalent resistance is the one at the terminals of C of the network where all sources have been zeroed.

9. I.1. Exercise. E1.3 The switch in the network shows in figure is open, in t=0 it is closed. Find the power P* in t*. The power is equal to the voltage*current. We have to calculate the current in R3 and the voltage across R3. They are both network variables that have to be evaluated in 0- , 0+ , +. After we have to calculate the time constant of the network and voltage and current at the time t*.

10. I.1. Exercise.

11. I.1. Exercise.

12. I.1. Exercise. -0.6 4.5 4.5 t Other method:

13. I.1. Exercise. E1.4 The switch in the network shows in figure is open, in t=0 it is closed. Find the current in the inductor from 0- to + for. The current in the inductor is a state variable than the networks in 0-, + and the time constant must be calculated. The different networks can be resolved calculating the Thevenin circuit equivalent at the PQ terminals. =0

14. I.1. Exercise. The networks that can be solved is shown in figure. The current flowing in the inductor from 0- to +is:

15. I.1. Exercise. E1.5 The switch in the network shows in figure is open, in t=0 it is closed. Find the voltage across the inductor from 0- to +. The voltage across L is a network variable than the networks in 0- , 0+ , + and the time constant must be calculated. R/2 2 0.75

16. I.1. Exercise. E1.6 The switch in the network shows in figure is open, in t=0 it is closed. Find the energy stored in the circuit at t*=2ms. The energy is store only in the inductance L. The current in the inductance is a state variable than the networks in 0- , + and the time constant must be calculated. J

17. I.1. Exercise. E1.7 The switch in the network shows in figure is open, in t=0 it is closed. Find the current i2 and i3 from 0- to +.

18. I.2. Exercise. E2.1 The switch in the network shows in figure is open, in t=0 it is closed. Find the energy stored in the circuit at t*. The energy depends on the current in the inductor and the voltage across capacitor, both are state variables than the networks in 0-, + and the time constants must be calculated (one for L and one for C). The network is of the type in figures, it can be decoupled in two parts from t>0.

19. I.2. Exercise. t=0- t=+

20. I.2. Exercise. Timeconstants J J

21. I.2. Exercise. E2.2 The switch in the network shows in figure is open, in t=0 it is closed. Find the energy stored in the circuit at t*=0,2ms and the power generated by E2. The energy depends the voltage across capacitor, the power in E2 depends on its current. The first is a state variables, the second a network variables. The network can be decoupled in two parts from t>0.

22. I.2. Exercise. t=0- t=+ t=0+ t

23. I.2. Exercise. E2.3 The switch in the network shows in figure is close, in t=0 it is opened. Find the energy stored in the circuit at t*. The energy depends on the current in the inductor and the voltage across capacitor, both are state variables than the networks in 0-, + and the time constants must be calculated (one for L and one for C). The network is of the type in figures, it can be decoupled in two parts from t>0.

24. I.2. Exercise. t=0- t=+

25. I.2. Exercise. t